SELBERG TRACE FORMULA 15

discrete series; and can be made to vanish by taking an appropriate kind of

imaginary part, to be denoted by Jm (see [Z2], § 2(11); the notation "7m" is

not used there, however). Formula (1-3) then follows from (1.6). Finally,

(1.2) is a straightforward consequence of the spectral decomposition.

Two main applications of the trace formulae are generalized Veyl laws and

prime geodesic theorems. The Veyl laws for the compact case state

([Z3], § 5):

def

(1.7) Nr(r,T) = E (Op(r)u., u.) « T/ln T

kj|T 3 J (s,m)

(where (s,m) are as usual the (fl,W) parameters of a).

The prime geodesic theorems for the compact case state ([Z2], § 4):

(1.8) *r(«r,T) ^

1 (

^

T

J

7 O

T

= JLop^u,, u ^ ^ e ( K T

+

0S)m(Te3/4T).

Here 7v,s,mare certain explicit constants ([Z2], § 3) which will be described

x

further in § 6.

The Veyl laws (1.7) are required in the proof of the prime geodesic

theorems (1.8). Also required is an estimate in the absolute sums

def

(1.9) |Nr(«r,T)| = S |0p(r)u-, u-|

|rj|T J J

The obvious estimate